Understandings which grow and evolve towards more

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understandings, which grow and evolve towards more complete meanings; d) most of the learning takes place through dialogues between the students themselves and in conversations with the teacher; and e) sophistication in students’ understanding of data develops within each point of view (local and global) and within the dynamic and flexible integration of those views. This study confirms that even if students initially do not make more than partial sense of their data analysis tasks, through the support of appropriate teacher guidance, class discussions, peer work and interactions, and ongoing cycles of experiences with realistic problem situations, students slowly build meanings and develop experts’ points of view on local-global approaches to data and data representations. REFERENCES Ben-Zvi, D. & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representations. Educational Studies in Mathematics, 45 , 35–65. Resnick, L. (1988). Treating mathematics as an ill-structured discipline. In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32-60). Reston, VA: NCTM. DANI BEN-ZVI University of Haifa, Faculty of Education Mt. Carmel, Haifa 31905 Israel.
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35 5. FROM DATA VIA ‘BUMP’ TO DISTRIBUTION ARTHUR BAKKER Utrecht University, The Netherlands [email protected] Students tend to see data as individual values and find it hard to reason with data sets as a whole that has certain characteristics such as an average representing the group, a majority and outliers, or a constant shape. An end goal of our teaching experiments was therefore that students could reason with distributions on an intuitive level in relation to shape (hill, bump). For this classroom-based research in six seventh-grade classes, the so-called statistical minitools initially designed by Cobb, Gravemeijer, and colleagues of the Vanderbilt University (Cobb et al 1997) were used. At the presentation, three episodes were discussed. The first video fragment showed how students found the mean visually in a value bar graph by a compensation strategy (figure 1). Student: “I cut off the long bits and give them to the shorter ones” . The presenter argued in the discussion that this representation is more suitable for developing understanding of the mean than the balance model. Figure 1. Compensation strategy on battery life span data in hours. The second video fragment was an example of what the presenter calls ‘extended sampling’ or ‘growing samples’. Students investigated sample size starting with four data points and adding new data to it up to 67 data points with dot plots. After that they predicted the shape of still larger samples. Their predictions were smoother and more hill-like than the investigated dot plots from smaller samples. In the last episode, students started to reason with ‘bumps’ after student-made graphs had been discussed. They predicted shapes and used terms like majority, outliers, sample size to explain their predictions.
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